MPI-PhT/94-19 TUM-T31-59/94

We point out that the measurement of just the two branching fractions B(K → πνν̄) and B(KL → πνν̄) can in a theoretically clean manner determine sin 2β almost independently of mt and Vcb. This allows to obtain an interesting relation between the CP asymmetry ACP (ψKS) in B physics and the branching ratios for these two rare K decays. The recently calculated next-toleading order QCD corrections improve the accuracy of this analysis. We find typically ∆ sin 2β = ±0.11 provided B(K → πνν̄) and B(KL → πνν̄) are measured within ±10% accuracy. With decreasing uncertainty in ΛMS and mc this error could be reduced to ∆ sin 2β < ±0.10. The determination of sin 2α and sin 2γ on the other hand is rather poor. However respectable determinations of the Wolfenstein parameter η and of | Vtd | can be obtained. ∗Supported by the German Bundesministerium für Forschung und Technologie under contract 06 TM 732 and by the CEC science project SC1–CT91–0729. The search for the unitarity triangle is one of the important targets of contemporary particle physics. It is well known that CP asymmetries in B → f , where f is a CP eigenstate will play an important role in this enterprise. The CP asymmetry in the decay B d → ψKS allows in the standard model a direct measurement of the angle β in the unitarity triangle without any theoretical uncertainties [1]. Similarly the decay B d → ππ gives the angle α, although in this case strategies involving other channels are necessary in order to remove hadronic uncertainties related to penguin contributions [2, 3, 4, 5, 6]. The determination of the angle γ from CP asymmetries in neutral B-decays is more difficult but not impossible [7, 8]. On the other hand it has been pointed out in [9] that measurements of B(K → πνν̄) and B(KL → πνν̄) could also determine the unitarity triangle completely provided mt and Vcb are known. It should be stressed that these two decays are theoretically very clean. KL → πνν̄ is dominated by short distance loop diagrams involving the top quark and proceeds almost entirely through direct CP violation [10]. K → πνν̄ is CP conserving and receives contributions from both internal top and charm exchanges. The recent calculation of next-to-leading QCD corrections to these decays [11, 12, 13] considerably reduced the theoretical uncertainty due to the choice of the renormalization scales present in the leading order expressions [14]. Since the relevant hadronic matrix elements of the weak current s̄γμ(1− γ5)d can be measured in the leading decay K → πeν, the resulting theoretical expressions for B(K → πνν̄) and B(KL → πνν̄) are only functions of the CKM parameters, the QCD scale ΛMS and the quark masses mt and mc. The long distance contributions to K + → πνν̄ have been considered in [15, 16, 17] and found to be very small: two to three orders of magnitude smaller than the short distance contribution at the level of the branching ratio. The long distance contributions to KL → πνν̄ are negligible as well. In view of this theoretical development and anticipating improvements in the knowledge of Vcb andmt we would like to make a closer look on this determination of the unitarity triangle. In particular we want to point out that sin 2β can be entirely expressed in terms of B(K → πνν̄) and B(KL → πνν̄) with only a very weak dependence on Vcb andmt. Consequently K + → πνν̄ andKL → πνν̄ offer an independent clean determination of sin 2β which can be confronted with the one possible in B → ψKS. Combining these two ways of determining